Theory of Geometrized Numbers: Sphere-Numbers and Their Algebra

The present manuscript describes a recent mathematical theory in which the traditional notion of quantity extends from its apex in order to a geometrical shape—asphere with a fixed radius in an n-dimensional space. Such objects, named sphere numbers, are distinguished by their dimensions and radius. The theory specifies important ways of adding, multiplying, and combining sphere numbers of the same or different dimensions.The framework is naturally connected to the well-known quantity systems—real numbers, complex numbers, four, and octonions—in dimension 1, 2, 4, and 8, as well as to all other dimensions. Calculus of function over sphere numbers, including differentiation and integration. The skeleton provides new tools for managing multivalent limits, symmetry-related problems, and interchanging between objects with different dimensions, as well as for physics and geometry.The paper develops an axiomatic base, research algebraic frameworks, and reasoning together with many unexploited difficulties to guide further research.